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Showing 18 of 280 problems in Partial differential equations

GEO-TOP-013

Carath\u00e9odory Umbilic Conjecture

MajorConjecture1924

Every closed

C^{2}

, strictly convex surface embedded in

R^{3}

has at least two umbilic points, where the two principal curvatures are equal.

ANAL-FA-009

Diagonal Fourier extension conjecture for compact paraboloids

MajorConjecturec. 1967

Let

d \geq 3

, let

P^{d - 1} = {(ξ, ∣ ξ ∣^{2}) : ξ \in R^{d - 1}} \subset R^{d}

, and let

d σ = η d S

, where

d S

is surface measure and

0 \leq η \in C_{c}^{\infty} (P^{d - 1})

is nonzero. Define

E_{σ} f (x) = \int_{P^{d - 1}} e^{2 πi x \cdot ω} f (ω) d σ (ω) .

For every

q > 2 d / (d - 1)

there is a constant

C = C (d, q, σ)

such that

∥ E_{σ} f ∥_{L^{q} (R^{d})} \leq C ∥ f ∥_{L^{q} (d σ)}

for every

f \in L^{q} (d σ)

PDE-001

Three-dimensional Navier–Stokes existence and smoothness

IconicOpen problem1934

Let

u_{0} : R^{3} \to R^{3}

C^{\infty}

, divergence-free, and satisfy

sup_{x} (1 + ∣ x ∣)^{N} ∣ \partial^{α} u_{0} (x) ∣ < \infty

for every

N

and multi-index

α

. There exist

u \in C^{\infty} (R^{3} \times [0, \infty); R^{3})

and

p \in C^{\infty} (R^{3} \times [0, \infty))

satisfying

\partial_{t} u + (u \cdot \nabla) u = - \nabla p + Δ u, \nabla \cdot u = 0, u (\cdot, 0) = u_{0}

for all

t \geq 0

, with

sup_{0 \leq t \leq T} \int_{R^{3}} ∣ u (x, t) ∣^{2} d x < \infty

for every

T < \infty

PDE-007

Weak cosmic censorship (future-null-infinity form)

IconicConjecture1969

Let

D_{AF}

be the space of smooth, complete vacuum initial data

(h, K)

R^{3}

satisfying

R_{h} - ∣ K ∣_{h}^{2} + (tr_{h} K)^{2} = 0

and

div_{h} K - d (tr_{h} K) = 0

. In fixed asymptotic coordinates, put

r = ∣ x ∣

and require

\partial^{α} (h_{ij} - δ_{ij}) = O (r^{- 1 - ∣ α ∣})

and

\partial^{α} K_{ij} = O (r^{- 2 - ∣ α ∣})

for every multi-index

α

. Give

D_{AF}

the relative weighted

C^{\infty}

topology induced by the seminorms

q_{m} (γ, L) = ∣ α ∣ \leq m \sum x sup (1 + r)^{1 + ∣ α ∣} ∣ \partial^{α} γ ∣ + ∣ α ∣ \leq m \sum x sup (1 + r)^{2 + ∣ α ∣} ∣ \partial^{α} L ∣

on differences

(γ, L) = (h - h^{'}, K - K^{'})

. There is an open dense subset of

D_{AF}

whose maximal globally hyperbolic developments have a conformal completion with complete future null infinity

I^{+}

, meaning that every physical null geodesic ending at

I^{+}

has infinite affine length.

PDE-002

Global regularity for three-dimensional incompressible Euler

LandmarkOpen problemc. 1930

For every divergence-free Schwartz vector field

u_{0} \in S (R^{3}; R^{3})

, the maximal classical solution of

\partial_{t} u + (u \cdot \nabla) u = - \nabla p, \nabla \cdot u = 0, u (\cdot, 0) = u_{0}

exists for all

t \geq 0

and belongs to

C^{\infty} (R^{3} \times [0, \infty))

PDE-003

Global regularity for inviscid surface quasi-geostrophic flow

LandmarkOpen problem1994

For every

θ_{0} \in S (R^{2})

, the unique maximal classical solution of

\partial_{t} θ + u \cdot \nabla θ = 0, u = \nabla^{⊥} (- Δ)^{- 1/2} θ = (- R_{2} θ, R_{1} θ), θ (\cdot, 0) = θ_{0}

exists for every

t \geq 0

and is smooth, where

R_{j}

is the

j

-th Riesz transform.

PDE-008

$C^2$ strong cosmic censorship

LandmarkConjecturec. 1979

Fix a closed smooth

3

-manifold

Σ

and a smooth background Riemannian metric

b

. Let

D_{vac} (Σ)

be the space of smooth pairs

(h, K)

, with

h

Riemannian and

K

symmetric, satisfying

R_{h} - ∣ K ∣_{h}^{2} + (tr_{h} K)^{2} = 0

and

div_{h} K - d (tr_{h} K) = 0

. Give it the relative Fréchet topology induced by

q_{m} (γ, L) = ∥ γ ∥_{C^{m} (b)} + ∥ L ∥_{C^{m} (b)}

m = 0, 1, \dots

, on differences

(γ, L) = (h - h^{'}, K - K^{'})

. In every nonempty connected component of

D_{vac} (Σ)

, the data whose maximal globally hyperbolic development admits no proper isometric embedding into a connected Lorentzian manifold with

C^{2}

metric form a residual set, meaning a countable intersection of open dense sets.

PDE-009

De Giorgi conjecture in dimensions $4$ through $8$

LandmarkConjecture1978

Let

n \in {4, 5, 6, 7, 8}

. If

u \in C^{2} (R^{n})

is bounded, satisfies

Δ u = u^{3} - u and \partial_{x_{n}} u > 0 on R^{n},

then

u

is one-dimensional: there are

ν \in S^{n - 1}

and

h \in C^{2} (R)

such that

u (x) = h (ν \cdot x)

for every

x \in R^{n}

PDE-010

Nonlinear Stability of Subextremal Kerr Black Holes

LandmarkConjecturec. 1965

For every fixed Kerr exterior with mass

m > 0

and angular momentum parameter

∣ a ∣ < m

, all sufficiently small asymptotically flat vacuum perturbations of compatible initial data have a complete exterior development that, after gauge adjustment, remains close to and asymptotically approaches a nearby Kerr exterior.

PDE-004

Global regularity for three-dimensional incompressible MHD

MajorOpen problemc. 1950

For every pair of smooth divergence-free vector fields

u_{0}, b_{0} : T^{3} \to R^{3}

, the unique local smooth solution of

\partial_{t} u + (u \cdot \nabla) u - (b \cdot \nabla) b + \nabla p \partial_{t} b + (u \cdot \nabla) b - (b \cdot \nabla) u \nabla \cdot u = \nabla \cdot b (u, b) ∣_{t = 0} = Δ u, = Δ b, = 0, = (u_{0}, b_{0})

extends smoothly for all

t \geq 0

PDE-005

Global regularity for the two-dimensional inviscid Boussinesq system

MajorOpen problemc. 1980

For every

u_{0} \in S (R^{2}; R^{2})

and

θ_{0} \in S (R^{2})

with

\nabla \cdot u_{0} = 0

, the maximal classical solution of

\partial_{t} u + (u \cdot \nabla) u + \nabla p = θ e_{2}, \partial_{t} θ + u \cdot \nabla θ = 0, \nabla \cdot u = 0, (u, θ) ∣_{t = 0} = (u_{0}, θ_{0})

exists for all

t \geq 0

and is smooth.

PDE-006

Global classical solutions of the relativistic Vlasov–Maxwell system

MajorOpen problemc. 1980

Let

f_{0}^{\pm} \in C_{c}^{\infty} (R_{x}^{3} \times R_{p}^{3})

be nonnegative and let

E_{0}, B_{0} \in C_{c}^{\infty} (R^{3}; R^{3})

satisfy

\nabla \cdot E_{0} = \int (f_{0}^{+} - f_{0}^{-}) d p

and

\nabla \cdot B_{0} = 0

. With

v (p) = p / 1 + ∣ p ∣^{2}

, the system

\partial_{t} f^{\pm} + v \cdot \nabla_{x} f^{\pm} \pm (E + v \times B) \cdot \nabla_{p} f^{\pm} = 0, \partial_{t} E - \nabla \times B = - j, \partial_{t} B + \nabla \times E = 0, \nabla \cdot E = ρ, \nabla \cdot B = 0, ρ = \int_{R^{3}} (f^{+} - f^{-}) d p, j = \int_{R^{3}} v (p) (f^{+} - f^{-}) d p

has a unique classical solution for all

t \geq 0

DYN-001

Uniform finiteness in Hilbert's sixteenth problem

IconicOpen problem1900

For every integer

d \geq 2

, there is an integer

H (d) < \infty

such that every planar polynomial vector field

\overset{x}{˙} = P (x, y), \overset{y}{˙} = Q (x, y), max {de g P, de g Q} \leq d,

having only finitely many limit cycles has at most

H (d)

limit cycles. A limit cycle is an isolated periodic orbit.

MATH-PHYS-009

General spacetime Penrose inequality

LandmarkConjecture1973

Let

(M^{3}, g, k)

be a smooth, connected, orientable, complete asymptotically flat initial data set. Define its energy and momentum densities by

16 π μ = R_{g} + (tr_{g} k)^{2} - ∣ k ∣_{g}^{2}, 8 π J = div_{g} (k - (tr_{g} k) g),

assume the dominant energy condition

μ \geq ∣ J ∣_{g}

, and fix an asymptotically flat end. Let

Σ

be an outermost apparent horizon relative to that end, allowing a disjoint union of marginally outer trapped components

H + tr_{Σ} k = 0

and marginally inner trapped components

H - tr_{Σ} k = 0

. Define

A

to be the infimum of the total

g

-areas of smooth closed surfaces enclosing

Σ

relative to the chosen end. If the ADM energy-momentum of the end is

(E, P)

and

m_{ADM} = E^{2} - ∣ P ∣^{2}

, then

m_{ADM} \geq \frac{A}{16 π} .

Equality should occur only when the exterior data arise from a spacelike slice of the Schwarzschild spacetime.

MATH-PHYS-010

Homogeneous interacting Bose-gas condensation

MajorConjecture1925

Let

v \geq 0

be a nonzero radial finite-range potential on

R^{3}

with scattering length

a

, let

Λ_{L} = (R / L Z)^{3}

, and let

v_{L}

be the

L

-periodic extension of

v

. For a normalized bosonic ground state of

H_{N, L} = - i = 1 \sum N Δ_{i} + i < j \sum v_{L} (x_{i} - x_{j}) on L_{sym}^{2} (Λ_{L}^{N}),

let

γ_{N, L}^{(1)}

be its one-particle density matrix, normalized by

Tr γ_{N, L}^{(1)} = N

. Then

ρ a^{3} ↓ 0 lim N, L \to \infty N / L^{3} = ρ lim \frac{λ _{m a x} ( γ _{N, L}^{(1)} )}{N} = 1.